
An interesting instance is, for example, the case where \(W = B^H\)  the fractional Brownian motion with Hurst parameter \(H\in (0,1)\)  and covariance
\(R_{B^H}(t,s) = \mathbb {E}[B^H_tB^H_s] = \frac{1}{2}\left( t^{2H}+s^{2H} - |t-s|^{2H}\right).\) 

For \(W_0 =0\)  (and \(R_W(0,0) = 0\) ), we have, by [1]},
\(\langle h_1,h_2\rangle _{\mathcal {H}} = \int _0^1\int _0^1 h_1(t)h_2(s)\mathop {}\!\mathrm {d}R_W(t,s),\) 
whenever the 2d Young integral on the right-hand-side is well-defined.


